Ordinary Mixed Life Insurance and Mortality-Linked Insurance Contracts M.Sghairi M.Kouki February 16, 2007 Abstract Ordinary mixed life insurance is a mix between temporary deathinsurance and pure endowment. Its duration is at least T years or more and in case of death the capital/pension will be paid to the inheritors and if the insured are alive, they will receive the payment. Thus, a single premium is paid by the policy-holders and placed by the insurance company on the incomplete financial market; but the difficulty lies in a fair tariffing. Our method is based primarily on two principles: equivalent martingale measure and the equality between the retrospective and the prospective market reserves. Then we obtain a condition on the discounting of the investment rate. This result generalizes the results obtained by M.Dahl [1] for a life insurance contract, in case of survival. Keywords: Life insurance; Stochastic mortality intensity; Incomplete financial markets; Lévy process; Equivalent measure principle 1 Introduction Traditionally, actuaries calculate premiums and market reserves using a deterministic mortality intensity that depends on age only, and a constant interest rate. But, in reality, neither the interest rate nor the mortality intensity are deterministic. It s well known, that we deal with two deferents life insurance contract : pure endowment and temporary death-insurance contracts. In this paper we address the case of an ordinary mixed insurance contract for only one x-year-old policy-holder, with stochastic mortality intensity, and an investment on the incomplete financial market. The ordinary 1
mixed insurance contract is a mix between temporary death insurance and pure endowment: - upon the policy-holder s death, a capital C0, 1 is paid to a designated recipient in the form of l benefits, in the event of death before the expiration date T of the contract, - a capital KT 2 is paid in full to the ensured person in the event of life at the expiry date (T) of the contract. This contract is paid by a single premium Π l 0 at time 0. In this paper we address the problem of fair pricing, which will be solved in two stages. First, by changing the initial probability measure in an equivalent measure, and then by expecting the retrospective and prospective market reserves. Thus, we obtain a necessary condition to the equity contract that relies on the discounting of the accumulation rate. In section 2 we present related to the financial market and those related the stochastic mortality. In section 3, we solve the problem by equalizing the time T retrospective and prospective market reserves. In section 4 we provide same Monte Carlo simulation experience, and section 5 concludes. 2 Assumptions and notations 2.1 The financial market We work on a space of probability (Ω, F,P) where filtration F = (F t ) t describes the information available on the incomplete financial market. We consider a financial market consisting of two traded assets : a risky asset with price process U and a locally risk-free asset with price process B (see T.Chan [4]). The stochastic process (U t ) t describes the accumulation rate. The dynamics of U and B may be given, for example, by : du t = α U (t,u t ) U t dt + σ U (t,u t ) U t dy t, > 0 db t = r t B t dt, B 0 = 1 (2.1) where r t 0 is the risk-free interest rate, α U (t,u t ) describes the expected behavior of the asset s instantaneous price, σ U (t,u t ) is uniformly bounded away from 0. It describes the sensitivity of the instantaneous price of the assets compared to the general trends in the 2
M.Sghairi, M.Kouki financial market. and (Y t ) t, a Lévy process on [0,T], is the index which describes the general trends in the financial market. For analytical purpose, we introduce the probability measure Q, equivalent to P, under which the discounted price process of the asset is a Q-martingale. Indeed, the expected discounted value is given by : p(t,t) = E Q [e R T t r udu F t ] so the financial annuity (see R.Cobbaut [3]) of an unity, paid in l periods, is given by l 1 a l = p(0,u). 2.2 The mortality u=0 The mortality is the second random component of the development of the ordinary mixed life insurance contract. It is linked to an F-adapted rightcontinuous Markov process Z = (Z t ) t [0,T], on a finite state space J = {0, 1} which describes the state of the insured person : 0 = alive, 1 = dead (see M.Dahl [1]). Only the transition from state 0 to state 1 is possible. On the one hand, under the initial probability P, the intensity of transition of Z is simply given at time t by the mortality intensity µ [x]+t, where, (µ [x]+t ) t is an adapted stochastic process. On the other hand, under the probability Q, the intensity of transition is multiplied by (1 + g t ), where (g t ) t is an adapted process, strictly higher than 1 Q-almost surely. Then we introduce the survival probability of an age [x]+t person from time t to T. It is given by G Q (t,µ [x]+t,t) = E Q [e R T t (1+gu)µ [x]+udu µ [x]+t ] where E Q indicates the expectation under Q. Then the death probability estimated at time t under Q is given by Ḡ Q (t,µ [x]+t,t) = 1 G Q (t,µ [x]+t,t). We denote by (K 2 t ) t T the adapted stochastic process, equal to the ensured sum at time t for pure endowment in the event of survival. In particular K 2 T indicates the sum to be paid at time T in the survival event at age x + T. 3
3 Solving the problem The equivalent measure Q principle yields the premium Π l 0 for the ordinary mixed life insurance contract, with sum C0 1 as the ensured sum in case of death, and KT 2 as the ensured sum in case of life. Let us notice that on model is an extension of the Norberg [2] model. Thus, the expected discounted value under Q of the actual payments having to be of null Q-mean, we have T Π l 0 = K0p(0,T)G 2 Q (0,µ [x],t) C1 0 l a l p(0,s) s G Q (0,µ [x],s)ds. (3.1) 0 Note that K 1,l 0 = C1 0 l a l the actual annuity value in l terms and the instantaneous risk of death is s G Q (0,µ [x],s). Proposition 3.1: Let the actual payments Π l 0 by given by (3.1) Then E Q [exp( T r 0 udu) U T ] = 1 + K1,l 0 [p(0,t)ḡq (0,µ [x],t) Π l 0 + T 0 (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds] (3.2) Proof : Let V i,l,retro t be the retrospective reserves conditioned at time t [0, T], given that the policy-holder is in the Z t = i state, is defined by: V i,l,retro t = Π l 0 U t K 1,l 0 1 {i=1} 1 {t<t } K 2 T 1 {i=0} 1 {t=t } (3.3) where K 1,l 0 = C1 0 l a l indicates the actual annuity value in l terms and i is person s state at time t. Let V i,l,pro t be the prospective reserves at time t, given that the policy-holder is in the Z t = i state. The sum insured in the event of life is Kt 2 and we consider that the sum insured in the event of death is constant in a course 4
M.Sghairi, M.Kouki of time equal to K 1,l 0. Therefore V i,l,pro t = E Q [K 1,l T 0 (1 + g t s)µ [x]+s e R s t ru+(1+gu)µ [x]+udu ds + e R T t r udu e R T 0 (1+gu)µ[x]+udu Kt 2 Z t = i, I t G t ], 0 t < T (3.4) To determine the adapted advantage, Kt 2, we use the following criterion: E Q [V Zt,l,pro t I t G t ] = E Q [V Zt,l,retro t I t G t ] (3.5) It is to be noted that in the above equality, the expectation relates only to the values of Z t. For the contract to be fair the expected discounted value under Q of the actual payments should be 0, i.e. E Q [Π l 0 + K 1,l 0 T 0 (1 + g s)µ [x]+s e R s t ru+(1+gu)µ [x]+udu ds K 2 T e R T 0 ru+(1+gu)µ [x]+udu ] = 0. (3.6) This leads to a condition on the process of accumulation U. Let us express K 2 T in terms of U, by the fact that the prospective reserve at time T must be null: V 0,l,pro T = V 1,l,pro T = 0. According to (3.3) we have : V 0,l,retro T = Π l U T 0 KT 2 K 1,l 0. Thus, criterion (3.5), applied at time T, gives : et V 1,l,retro T = Π l 0 U T KT 2 = Π l U T 1 0 G Q (0,µ [x],t) Ḡ Q (0,µ [x],t) K1,l 0 G Q (0,µ [x],t) with ḠQ (0,µ [x],t) = 1 G Q (0,µ [x],t) as the death probability from time 0 to time T. While inserting this in (3.6), one notes that the process U must check : E Q [e R T 0 rudu U T ] = 1 + K1,l 0 Π l [p(0,t)ḡq (0,µ [x],t) 0 + T 0 (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds]. (3.7) 5
In particular, it is to be noted that the expected discounted value of accumulation factor U T from 0 to T, E Q [exp( T r 0 udu) U T ], is higher than 1 whereas under the modeling of M.Dahl [1] it is equal to 1. After solving the problem of a fair contract we addressed the development of the benefits. Criterion (3.5) at time t < T is written as : Π l 0 Ut = e R t 0 (1+gu)µ [x]+udu K 2 t p(t,t)g Q (t,µ [x]+t,t) + K 1,l 0 T t (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds (3.8) Using the expression of premium (3.1), we express the advantages at time t according to the advantages at time 0 R t0 Kt 2 (1+gu)µ e [x]+u du = p(t,t)g Q (t,µ [x]+t,t) [K2 0p(0,T)G Q (0,µ [x],t) Ut + K 1,l 0 [ Ut T 0 (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds (3.9) T t (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds]. 4 Monte Carlo study In this section we simulate the loss risk. To get a realistic portfolio of the insured, it would be necessary to introduce policy-holders of different ages/cohorts, different horizons, and different dates of subscription. However, in this study, we started with simulations of homogeneous portfolios. We simulated the risk of loss of 1000 ordinary mixed life insurance contracts, concerning x-year insured people, x varying in somme interval I, and of fixed horizon T years. We used the mortality intensity of Gompertz-Makeham µ x = a + bc x of parameter a = 0.000233, b = 0.0000658 et c = 1.0959. Example: If the insured have different ages/cohorts, then the loss risk of the associated portfolio at T can be obtained as a weighted mean of the individuel risks. The left figure represent the dynamics of assets price is of the form du t = αu t dt + σu t dy t, where α = 0.6, and σ = 0.2, and Y t is a Lévy process of the 6
M.Sghairi, M.Kouki form dy t = dw t + dn t where W is the Brownian motion and N the Poisson process of 10% intensity, and that him of right-hand side the dynamics of assets price is of the form du t = αu t dt + σu t dw t. We calculated V Zt,x,retro t t [0,T], x I, given by formula (3.3) for 1000 contracts with C0 1 = 1, KT 2 = 0.5. We simulated the loser contract rate at time t [0,T] for N insured persons with ages x I using the approximation : t [0,T], x I : 1 N N i=1 1 {V x t <0} P(V x t < 0) Figure 1 : The left figure represent the risk of loss of contracts of 1000 insured persons, x [30, 60] years and horizon T = 30 years, with the dynamics of assets price is of the form du t = αu t dt + σu t dy t and that him of right-hand side, with the dynamics of assets price is of the form du t = αu t dt + σu t dw t. Figure 2 : The left figure represent the risk of loss of contracts of 1000 insured persons, x [40, 60] years and horizon T = 20 years, with the dynamics of assets price is of the form du t = αu t dt + σu t dy t and that him of right-hand side, with the dynamics of assets price is of the form du t = αu t dt + σu t dw t. Figure 3 :The left figure represent the risk of loss of contracts of 1000 insured persons, x [35, 55] years and horizon T = 20 years, with the dynamics of assets price is of the form du t = αu t dt + σu t dy t and that him of right-hand side, with the dynamics of assets price is of the form du t = αu t dt + σu t dw t. 7
Figure 1: Risk of loss Figure 2: Risk of loss 8
M.Sghairi, M.Kouki Figure 3: Risk of loss 5 Conclusion Our objective in this study was to find a tool to determine the premium that policy-holders have to pay to insurers according to the terms of the contracts. However the contracts mention the expiry date, as well as the mode of payment of the annuities due by the policy-holder and the two parties operate in a context of information asymmetry. In this context each individual tries to maximize his/her utility function, in other terms his/her future benefit. The uncertainty that prevails in all incomplete contracts requires an estimate of the risk induced by incomplete information. From this point of view we tried to design a mathematical model in order to help the insurer take his/her decision, estimate the risks, and determine the premium of the contract while taking into account unquestionable variables: mortality intensity rate, the characteristics of each insurance contract(temporary death-insurance, pure endowment). Certain assumptions were put forward to design a model that considers the determination of the premium as well as the risk. An empirical study was undertaken to highlight the reliability of our model. Indeed the model shows that in the event of non-diversification of the insurer s portfolios, the risk of loss caused varies according to the expiry date of the contract. However these results must not be considered as the limits of our model, since the data used to test this model were subjective data which do not reflect 9
the reality. Collecting data in this field remains complex and it is the major problem for the validation of our model. References [1] M. Dahl. Stochastic mortality in life insurance:market reserves and mortality-linked insurance contracts. Insurance:Mathematics and Economics 35., pages 113 136, 2004. [2] R. Norberg. Reserves in life and pension insurance. Scandinavian Actuarial Journal 1, pages 3 24, 1991. [3] R.Cobbaut. Théorie Financière. Economica, Paris, 1994. [4] T.Chan. Pricing contingent claims on stocks driven by lévy processes. The Annals of Applied Probability., pages 504 528, 1999, Vol.9. M.Sghairi M.Kouki Université PARIS DESCARTES Ecole Supérieure de la Statistique LABORATOIRE MAP5 et de l Analyse de l Information 45, RUE DES SAINTS-PÈRES 6, rue des mètiers 75270 PARIS CEDEX 6 2035 Charguia 2 Ariana FRANCE TUNISIE sghairi@math-info.univ-paris5.fr mokhtar.kouki@essai.rnu.tn 10